TITLE: Non Kähler blow-up of a Kähler manifold
QUESTION [5 upvotes]: Is it possible to find a complete, non compact Kahler manifold $(X,\omega)$ with a closed, connected, non compact complex submanifold $Y\subset X$ of codimension at least 2 such that the blow-up of $X$ along $Y$  is not Kähler?

REPLY [4 votes]: When $Y$ is compact, the blow-up is always Kahler;
see e.g. Lemma 3.4 in this paper
(this is a generally known folklore theorem which we 
had to use, and hence written down).
For $Y$ non-compact the argument should be similar, but more cumbersome.